Chapter 09

Rotation of Rigid Bodies

Introduction

Rotational motion is a fundamental concept in physics, describing how objects spin about a fixed axis. Examples include airplane propellers, revolving doors, ceiling fans, and Ferris wheels. In this chapter, we focus on the ideal case of rigid bodies, which do not deform during rotation.

• Rigid body: An object whose shape does not change during motion; all points maintain fixed distances from each other.

• Rotation axis: The straight line about which the body rotates.

• Real-world context: While actual objects may stretch or twist, the rigid body model simplifies analysis.

Angular Coordinate

The position of a rotating object is described by its angular coordinate, typically denoted as $\theta$. This is the angle between a reference direction (often the x-axis) and the line from the axis of rotation to a point on the object.

• Fixed axis rotation: The axis remains stationary while the object rotates around it.

• Example: A car's speedometer needle rotates about a fixed axis, and its position is specified by the angle $\theta$ from the reference direction.

Units of Angles

Angles are measured in radians or degrees, but radians are preferred in physics for mathematical convenience.

• Radian: The angle subtended at the center of a circle by an arc equal in length to the radius.

• Relationship: One complete revolution is $360^\circ = 2\pi$ radians.

• Formula: $\theta = \frac{s}{r}$, where $s$ is the arc length and $r$ is the radius.

Table: Angle Units Comparison

Unit	Definition	Conversion
Degree	1/360 of a full circle	$360^\circ = 2\pi$ radians
Radian	Arc length equals radius	$1$rad$\approx 57.3^\circ$

Angular Velocity

Angular velocity describes how fast an object rotates and in which direction. It is defined as the rate of change of angular coordinate with respect to time.

• Average angular velocity: $\omega_{avg} = \frac{\Delta \theta}{\Delta t}$

• Instantaneous angular velocity: $\omega = \frac{d\theta}{dt}$

• Direction: Specified by the right-hand rule; positive for counterclockwise, negative for clockwise rotation about the z-axis.

• Vector nature: Angular velocity is a vector, pointing along the axis of rotation.

Angular Acceleration

Angular acceleration measures how quickly the angular velocity changes. It is important for analyzing rotational dynamics, especially when the rotation rate is not constant.

• Average angular acceleration: $\alpha_{avg} = \frac{\Delta \omega}{\Delta t}$

• Instantaneous angular acceleration: $\alpha = \frac{d\omega}{dt}$

• Significance: Positive $\alpha$ means speeding up; negative $\alpha$ means slowing down.

Rotational Kinematics with Constant Angular Acceleration

When angular acceleration is constant, the equations of rotational motion mirror those of linear kinematics.

• Key equations:

  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

  • $\omega = \omega_0 + \alpha t$

  • $\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$

• Comparison: These equations are analogous to those for straight-line motion with constant acceleration.

Relating Linear and Angular Quantities

Points on a rotating object have both angular and linear motion. The relationship between these quantities depends on the distance from the axis of rotation.

• Linear speed: $v = r\omega$

• Tangential acceleration: $a_t = r\alpha$

• Centripetal (radial) acceleration: $a_c = \frac{v^2}{r} = r\omega^2$

• Important: Always use radians when relating linear and angular quantities.

Rotational Kinetic Energy

Rotating objects possess kinetic energy due to their motion. The rotational kinetic energy depends on the object's moment of inertia and angular velocity.

• Formula: $K_{rot} = \frac{1}{2} I \omega^2$

• Moment of inertia ($I$): A measure of how mass is distributed relative to the axis of rotation.

• SI unit: kilogram-meter squared ($\text{kg} \cdot \text{m}^2$)

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in rotational motion. It depends on both the mass and how far the mass is from the axis of rotation.

• Definition: $I = \sum m_i r_i^2$ for discrete masses, or $I = \int r^2 dm$ for continuous bodies.

• Examples:

  • Moving mass closer to the axis decreases $I$.

  • Moving mass farther from the axis increases $I$.

• Applications: Bird wings—hummingbirds have small $I$ (rapid flapping), condors have large $I$ (slow flapping).

Table: Moments of Inertia for Common Shapes

Object	Axis	Moment of Inertia ($I$)
Slender rod	Center	$\frac{1}{12} mL^2$
Slender rod	End	$\frac{1}{3} mL^2$
Rectangular plate	Center	$\frac{1}{12} m(a^2 + b^2)$
Rectangular plate	Edge	$\frac{1}{3} ma^2$
Solid cylinder	Central axis	$\frac{1}{2} mR^2$
Hollow cylinder	Central axis	$mR^2$
Solid sphere	Center	$\frac{2}{5} mR^2$
Thin-walled hollow sphere	Center	$\frac{2}{3} mR^2$

Gravitational Potential Energy of an Extended Object

The gravitational potential energy of an extended object can be calculated as if all its mass were concentrated at its center of mass.

• Formula: $U_{g} = M g y_{cm}$, where $y_{cm}$ is the vertical position of the center of mass.

• Application: High jumpers arch their bodies so the center of mass passes under the bar, minimizing the required energy.

Parallel-Axis Theorem

The parallel-axis theorem relates the moment of inertia about any axis to that about a parallel axis through the center of mass.

• Formula: $I = I_{cm} + M d^2$, where $d$ is the distance between axes.

• Application: Used to calculate $I$ for axes not passing through the center of mass, such as for a baseball bat swung about its end.

Moment of Inertia Calculations

For complex shapes, the moment of inertia is calculated by integrating over the object's volume.

• Formula: $I = \int r^2 \rho \, dV$, where $\rho$ is the mass density.

• Application: Geophysicists use satellite data to measure Earth's moment of inertia, revealing its internal mass distribution.

Additional info: The study notes have expanded on brief points and filled in missing context for clarity and completeness.